1877.] 



seen through a Crystalline Plate. 



397 



the primary plane) for points in the sections AC, BB' respectively. As 

 starts from zero, p starts from a and increases, and p x ' starts from 

 & _1 c 2 and decreases. "When becomes BOE, p x ' vanishes, and beyond that 

 becomes negative, while p continues to increase. As increases to 

 BOQ, p increases positively, and p^ negatively, to infinity, and beyond 

 that both change sign, p' becoming negative and p\' positive. As 

 increases to ANE, p decreases negatively to zero, while p x ' decreases 

 positively from infinity. On passing ANE, p becomes positive, and in- 

 creases to its final value, c, which it reaches when 0= 90°, while p x ' decreases 

 to its final value, b~ x a 2 . Thus though a>cwe may say that as increases 

 from to 90°, p' increases from a to c by passing through oo and 0, and p' 

 decreases from b~ l c 2 to b~ l a 2 by passing through and oo. 



The extravagant changes of apparent index in the immediate neigh- 

 bourhood of the wave- and ray-axes could probably not well be followed 

 by the microscope, on account of the necessity of working with pencils of 

 finite angular aperture, which would make the phenomena of focusing 

 blend themselves with those of conical refraction. But there can be little 

 doubt that a large increase or diminution of apparent index on approach- 

 ing the critical region would be readily discernible. That these changes 

 are not confined to the principal plane is evident, inasmuch as one prin- 

 cipal radius of curvature of the wave-surface becomes infinite at any point 

 of the circle of contact of the surface with the tangent plane perpendi- 

 cular to the optic axis, and one principal radius of curvature vanishes 

 at the conical point, to whatever normal section it be thought of as 

 belonging. 



Let us now resume the equations (2), (10), which give the principal 

 curvatures for the elliptic section, without deciding beforehand any thing 

 as to the relative magnitude of the parameters. 



As 6 changes from to 90°, the radius of curvature in the primary 

 plane changes from a~V to c _1 « 2 , and that in the secondary plane from 

 a to c ; and the ratio of the radii, therefore, changes from a~ 2 c 2 to c~V, 

 of which one is greater than 1 and the other less than 1. If, then, both 

 radii remain positive, as is the case in the two principal planes passing 

 through the mean axis, the two radii must be equal for some intermediate 

 value of 0. Hence there must be four umbilici in each of these planes. 

 To find the umbilici we must equate the values of p, p given by (2), (10), 

 whence 



a 2 c 2 {(b 2 -a 2 ) cos 2 0+(6 2 -c 2 ) sin 2 0}(cos 2 0+sin 2 0) 

 = { a 2 (b 2 - a 2 ) cos 2 + c\b 2 - c 2 ) sin 2 } (a 2 cos 2 + c 2 sin 2 0), 



which gives, after reduction, 



a\b 2 -a 2 ) 



This expression shows that the umbilici in the elliptic section made by 



